In Exercises 15–58, find each product. (x−3)²

Verified step by step guidance

1. Recognize that the expression \((x - 3)^2\) represents a binomial squared. This means you will expand it using the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify the terms in the binomial \((x - 3)\): here, \(a = x\) and \(b = 3\).
3. Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\) to the given expression. Substitute \(a = x\) and \(b = 3\) into the formula.
4. Simplify each term: \(a^2 = x^2\), \(-2ab = -2(x)(3) = -6x\), and \(b^2 = 3^2 = 9\).
5. Combine the simplified terms to write the expanded form of the expression: \(x^2 - 6x + 9\).

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. In this case, (x - 3)^2 can be expanded using the formula (a - b)^2 = a^2 - 2ab + b^2, where a = x and b = 3. Understanding this concept is essential for correctly applying the expansion to find the product.

Square of a Binomial

The square of a binomial is a specific case of binomial expansion where a binomial expression is multiplied by itself. For (x - 3)^2, this means multiplying (x - 3) by (x - 3). The result will yield a quadratic expression, which is a polynomial of degree two. Recognizing this pattern helps in simplifying and solving similar algebraic expressions.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form ax^2 + bx + c, where a, b, and c are constants, and a is not zero. The result of expanding (x - 3)^2 will yield a quadratic expression. Understanding the structure of quadratic expressions is crucial for further analysis, such as factoring, graphing, or solving equations derived from them.